"As a simple private FOSS user I would like to migrate my old 8.8.9 server to 8.8.10."If you really are a "simple private FOSS user", I would recommend to hire a qualified partner to do it for you. Also, do you really need to migrate or can't you just upgrade in place?

"I use licensed Zimbra Suite Plus to realtime backup my "old" server. What is the best approach?"I would recommend to you to directly buy ZeXtras Backup from ZeXtras, that way you make sure you get the support directly from the people developing the solution. Technically speaking, best approach is to perform an incremental migration with the zextras tool. Google it.

As long as you are staying with the same flavor of Linux, ie Ubuntu > Ubuntu or CentOS > CentOS, all you need is the /opt/zimbra directory and the ldap database, the exact same Zimbra version, and for the new server to have the same hostname and ip as the old. I've done this many times without issue.